A STUDY ON INTEGRAL TRANSFORMS OF THE GENERALIZED LOMMEL-WRIGHT FUNCTION Текст научной статьи по специальности «Математика»

ОРИГИНАЛНИ НАУЧНИ РАДОВИ ОРИГИНАЛЬНЫЕ НАУЧНЫЕ СТАТЬИ

ORIGINAL SCIENTIFIC PAPERS S

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A STUDY ON INTEGRAL TRANSFORMS OF THE GENERALIZED LOMMEL-WRIGHT FUNCTION

Mohammad Saeed Khana, Sirazul Haqb,

Moharram Ali Khanc, Nicola Fabianod |

Sefako Makgatho Health Sciences University, q

DOI: 10.5937/vojtehg70-36402;https://doi.org/10.5937/vojtehg70-36402 FIELD: Mathematics

ARTICLE TYPE: Original scientific paper Abstract:

Introduction/purpose: The aim of this article is to establish integral transforms of the generalized Lommel-Wright function.

Methods: These transforms are expressed in terms of the Wright Hyper-geometric function.

Results: Integrals involving the trigonometric, generalized Bessel function and the Struve functions are obtained.

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Department of Mathematics and Applied Mathematics,

Ga-Rankuwa, Republic of South Africa,

e-mail: [email protected], ro

ORCID iD: ©https://orcid.org/0000-0003-0216-241X b <u

b J. S. University, Department of Applied Science, ^

Shikohabad, Ferozabad, U.P., Republic of India,

e-mail: [email protected], o

ORCID iD: ©https://orcid.org/0000-0001-9297-2445

c Umaru Musa Yaradua University, o

Department of Mathematics and Statistics,

Katsina, Federal Republic of Nigeria,

e-mail: [email protected],

ORCID iD: ©https://orcid.org/0000-0002-5563-4743

d University of Belgrade, "Vinca" Institute of Nuclear Sciences -Institute of National Importance for the Republic of Serbia, Belgrade, Republic of Serbia,

e-mail: [email protected], corresponding author, w

ORCID iD: ©https://orcid.org/0000-0003-1645-2071

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Conclusions: Various interesting transforms as the consequence of this method are obtained.

Key words: Generalized Lommel-Wright functions J(z), Hankel transform, K-transform, Wright function, Whittaker function.

Introduction

The transform defined by the following integral equation

r

Rv{f (x);p} = g(p, v) = (px)l/2Kv(px)f (x)dx (1)

J 0

is called the k transform with p as a complex parameter and Kv(px) is called the Modified Bessel function of the third kind or the Macdonald function, see (Mathai et al, 2010, p.53). The Hankel transform of a function f (x), denoted by g(p, v) is defined as

f+m

g(p, v) = (px)1/2Jv(px)f (x)dx, p> 0 J 0

(2)

where Jv(px) is called the Bessel-Maitland function or the Maitland-Bessel function (Mathai et al, 2010, p.22 and p.56).

The Wright hypergeometric function defined by the series (Srivastava & Manocha, 1984):

p^q

(ai,Ai), .., (ap,Ap);

L (Pl,Bl).., (Pq ,Bq )

n r(aj + Ajk)zk

Ej=i_

_q '

k=0 Û r(Pj + Bjk)k! j=i

(3)

where the coefficients A1,....Ap and Bi,....Bq are positive real numbers such that

qp

(4)

1 + E Bj -E a ^0' j=1 j=1

can be slightly generalized (3) as given below.

p^q

(ai, 1), .., (ap, 1); (Pi, 1)..., (Pq, 1)

n r(aj) j=i

n r(Pj) j=i

pFq

ai,.., ap;

Pi,...,Pq

(5)

264

z

z

z

where pFq is the generalized hypergeometric function defined by (Srivas-tava & Manocha, 1984; Rainville, 1960)

F

p Fq

ßl,..,ßq

£

k=0

(ai)n,(ap)nzn (ßi)n,...., (ßq )n n!

pFq (ai,..., ap ; ßi, ....,ßq ; z),

(6)

2

3 6

2.

p. p

io cti

where (A)„ is the well known Pochhammer symbol (Srivastava & Manocha, 1984).

The series representation of the generalized Lommel Wright function as (Kachhia & Prajapati, 2016);

4;m(z) = E

k=0

(-i)k r(k + i)( § )2k+v+2A r(A + k + 1)mr(v + kß + A + 1)k! '

(z £ C/(-œ, 0], m £ N, v,A £ C,ß> 0).

(7)

Also, we have the following relations of the generalized Lommel Wright functions with trigonometric functions and the generalized Bessel function $V,\(Z) and the Struve function as follows:

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rm or

f s

g e

JilA0(z)= 'nzSin(z)

J l\\o0(z) = \ — cos(z) V nz

-1/2,0(

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y

d u t

s

A,

al

t e S.

Jßv'l (z) = KAz)

(10)

a h K

JV,'l/2(z) = Hv (z) •

(11)

The following known results of Mathai and Saxena (Mathai & Saxena, 1973):

r+<x

/ x5-1 Jn (ax)dx = 0

25-la-5 r( 5++n )

r(i +— )

, n(n) < K(5) < 3/2, a> 0 (12)

265

z

CM r+x>

0 / xS-1Kv(ax)dx = 2S-2a-Sr(S ± n)/2, (13)

J 0

o

P xi~' exp(-at)K„(ax)dx = (2a)^-T±(;)+/21/2) • (14)

- x- exp(1/2 xW„,(x)d* = - S), (15)

IT ZD O o

< O

x i xS-1 exp(-1/2 x)M(n,m)(x)dx

o Jo

LU

■ r(2m + 1)r(m + S + 1/2)r(n - S) (16)

dc r(m - S + 1/2)r(m + n + 1/2) ' (

neralizations and cases of the Lommel-Wright function have been investigated. For details, see (Paneva-Konovska, 2007; Menaria et

al, 2016; Mondal & Nisar, 2017; Srivastava & Daoust, 1969; Kiryakova, 2000).

§ Integral formulas involving the Lommel-Wright functions have been de-

ep veloped by many authors. See e.g., (Choi & Agarwal, 2013; Choi et al, 2014; Jain et al, 2016; Chaurasia & Pandey, 2010). In this sequel, here, we aim at establishing a certain new generalized integral formula involving the generalized Lommel-Wright function J^xZ interesting integral formulas which are derived as special cases.

Main results

This section deals with the evaluation of integrals formulas involving the Lommel-Wright function defined in (7) and the integrals involving the product of the Bessel function of first kind, the Kelvin's function and Whittaker function (Whittaker & Watson, 2013) with the generalized Lommel-Wright function.

Theorem 1. Let z e C/(-rc>, 0], m e N, v,A e C, i > 0. Then the Hankel transform of the generalized Lommel-Wright function defined in (7) is given by

1 ( b \ v+2A ( 2 ^ P+W(U+2X)

r° zp-ijn ww^ vz(a)

2^m+2

2\2j \a

(1, 1), ( V+P+Wv+2WX (A + 1,1),..., (A + 1,1), (v + A + 1,i), ((), -w) - b2 )( 4 x w ]

(18)

Proof. On using (7) in the integrand of (1) which is verified by uniform convergence of the involved series under the given conditions, we get

£

n=0

/ zp-1 Jv (az)J^m(bzW )dz = Jo '

(-1)nr(n + 1)(b/2)2n+V+2A f+« zP+w{2n+v+2X)-1

r(A + n + 1)mr(v + A + ni + 1)n! Jo

Jn (az)dz.

Now using (12) in the above equation we get

p+w(v+2A)

zp-1 J(azJm(bzw)dz = (1/2) (Vy ^2/a) r(n + 1)r(n + p + wv + 2wA + 2wn)/2^ - b2/^ ^4/a2^

E

n=o

r(A + n + 1)mr(v + A + ni + 1)r(2 + n - p - wv - 2wA - 2wn)/2 n!

1 ( b\ v+2X /^ p+w(v+2A)

2W W

- I X

(1,1)7 n+p+wv+2wX ,w

(A + 1,1),(A + 1,1), (v + A + 1,i), ((^v-^^2^ , -w^j ;

(W

(19)

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Theorem 2. Let z e C/(-&>, 0], m e N, v, A e C,i > 0. Then the K-Transform of the generalized Lommel-Wright function defined in (7) is given

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f+m 1 / b\ v+2x /

I zp-1Kv (azjm(bzw )dz = 4(jj

2^m+1

(1,1), (p+wv+22wX±n,w)

1 ( b\ V+2X ( 2\ P+W(v+2X)

4 V 2 / \a) '

4

(A + 1,1), •••, (A + 1,1), (v + A + 1,x); V 4

(20)

Proof. On using (7) in the integrand of (2) which is verified by uniform convergence of the involved series under the given conditions, we get

f+<X

/ Zp-1KV (az)4m(bzw )dz =

Jo '

JO

~ (-lyre+l)(b/2)^»> zP+w^+v+2x)-iKv(az)dz.

E

n=0

r(A + n + 1)"T(v + A + nf + 1)n! ,/0

Now using (13) in the above equation we get

v+2X ( \ p+w(v+2A)

zp-1Kv(azjm(bzw)dz = (1/4) (Vy ^2/a)

r(n + 1)r(p + wv + 2wA ± n +

E-

n=0

(\n/ \ nw

- b2/4) (4/a2)

r(A + n + 1)mr(v + A + nf + 1)n!

1 ( b\ v+2X f 2 \ P+w(v+2X)

= w

(1, 1), ( P+wv+22wX±V , w); / -tf W 4 X1

4 ) \a2)

+2wx±n, w); _ (A + 1,1), •••, (A + 1,1), (v + A + 1

(21)

□

Theorem 3. Let z e C/(-rc>, 0], m e N, v, A e C,x > 0. Then the K-Transform of the generalized Lommel-Wright function defined in (7) is given by

f+m

/ zp-1 exp(-az)Kn(az)J^m(b zw)dz =

Jo '

2av^ b/2

v+2X

(2a)P+w(v+2X)

2^m+2

(A + 1,1),

(1,1), (p + wv + 2wA ± n, 2w); (A + 1,1), (v + A + 1,f), (p + wv + 2wA + 1/2,2w);

x

o

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\4(4a2)_ J

r+<x

/ zp-1 exp(—az)K (az)J^m(bzw )dz J 0 '

(—1)nr(n + 1)(b/2)2n+v+2A

Now using (14) in the above equation we get

f+™ 2a^(b/2

0 z" eM—az)Kv(az)-vm(bzw)dz _

g r(n + 1)r(p + wv + 2w\ ± n + 2wn)()

n=0

r(A + n + 1)mr(v + A + nf + 1)(p + wv + 2wA + 1/2, 2w)n!

2^m+2

_ 2a^(b/2)v+2A

_ (2a)P+w(v+2A) x

(1,1), (p + wv + 2wA ± n, 2w); (A + 1,1),..., (A + 1,1), (v + A + 1,f), (p + wv + 2wA + 1/2,2w);

\4(4a2)w )

(24)

□

3

^m+1

(a)P+e(»+2X)r(1/2 ± a — n) (1,1), (1/2 ± a + p + vd + 2A9, 2d), (—n — p — vd — 2dA, —2d); (A + 1,1),..., (A + 1,1), (v + A + 1,f);

rn CO

<m

S± CP

Proof. On using (7) in the integrand of (3) which is verified by uniform convergence of the involved series under the given conditions, we get o

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^r(A + n + 1)mr(v + A + nx + 1)n!" |

n=0 t

o

zP+_(2n+v+2\)-1 exp(—az)Kv (az)dz. (23) |

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Theorem 4. Let z e C/(—<x, 0], m e N, v,A e C,x > 0. Then the pro- ^ duct of the Whittaker function and the generalized Lommel-Wright function defined in (7) is given by

( \ v+2\

rw/2 l zp-1 exp(az/2)WVa(az)J^m(w zd)dz _ +^+2^-

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Proof. Putting az = x, adz = dx as z ^ 0,x ^ 0 and z ^ x ^ and using (7) in the integrand of (4) which is verified by uniform convergence of the involved series under the given conditions, we get

r+<x

/ zp-1 exp(az/2)Wv,a (az)J^m(wz0 )dz = Jo ' '

v+2X

w/2 ) +(X

E

(ay+C+M) n=0

f

r(A + n + 1)mr(v + A + np + 1)n! xp+e(2n+u+2X)-1 exp(x/2)Wv,a(x)dx.

Now using (15) in the above equation we get

r(w /2) v+2X

l zp-1 eM.az/2)Wn,a(az)Jí;:r(wz°)dz = ay+e{vUT{1/2 ±

o

E

n=0 L

(a)P+e(v+2X'Y(1/2 ± a - n) r(n + 1)r(1/2 ± a + p + Ov + 20A + 26 n)r(-n - p - Ov - 20A - 29 n)

G(a2)0

r(A + n + 1)mr(v + A + np + 1)n!

v+2X

2 \ nl 2)0

w/2

(a)p+0(v+2X)r(1/2 ± a - n)

(1,1), (1/2 ± a + p + Ov + 2OA, 29), (-n - p - vO - 2OA, -29); (A + 1,1),..., (A + 1,1), (v + A + 1, p); ,2

(26)

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Theorem 5. Let z e C/(-rc>, 0], m e N, v,X e C,p > 0. Then the product of the Whittaker function and the generalized Lommel-Wright function defined in (7) is given by

r

/ z— exp(-az/2)MVa(az)J^m(w ze)dz = Jo ' '

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/ \ v+2X

(w/2\ (1/a)9(v+2X)r(2a + 1)

_-—-x

(a)^r(a + n + 1/2)

(1,1), (a + p + 1/2 + vd + 2Xd, 2d), (n - p - vd - 2d\, -2d); (X + 1,1),..., (A + 1,1), (v + A + 1, p), (a - p - dv - 2dX + 1/2, -2d)

V4(a2)V

(27)

Proof. Putting az = x, adz = dx as z ^ 0,x ^ 0 and z ^ +œ, x ^ and using (7) in the integrand of (5) which is verified by uniform convergence of the involved series under the given conditions, we get

f+m

/ zp-1 exp(-az/2)Mv , a (azJ m(wz9 )dz = J o ' '

w/2

v+2X

(1/a)9(v+2X)

(a)p

£

n=0 /■+<x

r(X + n + 1)mr(v + X + np + 1)n! xP+e(2n+v+2x)-1 exp(-x/2)Mn , a(x)dx.

Now using (16) in the above equation we get

E

n=0

r+<x

/ zp-1 exp(-az/2)Mv , a(az)J^ f(wze )dz = 0

(w/2)v+2X(1/a)e(v+2XX)Y(2a + 1) (a)PT(a + n + 1/2) X

r(n + 1)r(a + p + dv + 2dX + 1/2 + 2d n)r(n - p - dv - 2dX - 2d n) r(X + n + 1)mr(v + X + up + 1)r(a - p - dv - 2dX - 2nd + 1/2)n!

( )n] = (w/2) U(a2)V

v+2X

(1/a)9(v+2X)r(2a + 1)

(a)pr(a + n + 1/2) (1,1), (a + p + 1/2 + vd + 2Xd, 2d), (n - p - vd - 2dX, -2d); (X + 1,1),..., (X + 1,1), (v + X + 1, p), (a - p - dv - 2dX + 1/2, -2d)

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Theorem 6. Let z e C/(-rc>, 0], m e N, v,\ e C,p > 0. Then the product of the Whittaker function and the generalized Lommel-Wright function defined in (7) is given by

CC /X V+2X

E w/2 (1/a)0(v+2X

O zp-1 Wv,a (az) W-n,a (az) J^m(w z0)dz = ±-J -

0 Jo ' (a)p

0

- (1,1), (p+0(u+2X)+1 ± a, O), (p + 9(v + 2A) + 1,29);

(A + 1,1),..., (A + 1,1), (v + A + 1, p), 2(1 + p+0(v2+2X) ± n, O)

3^m+2

* ( -w2 \

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2

(29)

Proof. Putting az = x, adz = dx as z ^ 0,x ^ 0 and z ^ x ^ and using (7) in the integrand of (6) which is verified by uniform convergence of the involved series under the given conditions, we get

( \ v+2\

w/2 (1/a)°(v+2X) J zp-1W-v, a(az)Wv, a(az)J^ m(wzd )dz = ^-J--^-x

x Jo

§ r(n + 4-W)

O E

n=o

r(A + n + 1)mr(v + A + np + 1)n!

r+œ

/ Xp+d(2n+V+2X)-1W-n,a(x)Wv,a(x)dx. o

Now using (17) in the above equation we get

f+™ p-(w/2)v+2X(1/a)0(v+2X)

zp-1W-n,a(az)Wn,a(az)Jp;'Xm(wze )dz =

(a)p

r(n + 1)r(p+0(v+2¡X+2n+1) ± a)r(p + Ov + 2OA + 2O n)(~-Wïï)n n=0 r(A + n + 1)mr(v + A + np + 1)2r(1 + p+0(v+22X+2n) ± n)n!

(

X v+2X

w/2 J (1/a)0(v+2X) -x

(a)p

n

X

o

3^m+2

(1,1), (p+0(v+2X)+1 ± a, 0), (p + 0(v + 2A) + 1, 20); (A + 1,1),..., (A + 1,1), (v + A + 1, f), 2(1 + p+e(v+2X) ± n, 0);

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(30)

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Special cases

In this section, we get some integral formulas involving a trigonometric function and the generalized Lommel-Wright function as follows:

Corollary 1. If we take m = 1, f = 1, A = 0 and v = 1/2 in (1) and then by using (8), we derive the following integral formula:

zp-W/2-ijn (az) sin(b zw )dz = ^ a)

(*)

p+w/2

(

(3/2,1), ((

P+w/2

2+n-(p+w/2) 2

), -w);

-b2 4

(31)

Corollary 2. If we take m = 1, f = 1, A = 0 and v = 1/2 in (2) and then by using (8), we obtain:

jT+0° zp-w/2-1Kv(az) sin(b zw)dz = (W) a)

p+w/2

1^1

(

p+w/2±n

w); (4

(3/2,1);

4

(D

(32)

Corollary 3. If we take m = 1, f = 1, A = 0 and v = 1/2 in (3) and then by using (8), we obtain:

zp-w/2-1 exp(-az)Kn(az) sin(b zw)dz = j^)^)

0 1

; \4(4a2)wy

(p + w/2 ± n, 2w); (3/2,1), (p + w/2+ 1/2, 2w)

(33)

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Corollary 4. If we take m = 1, p = 1, A = 0 and v = 1/2 in (4) and then by using (8), we obtain:

r

/ zp-e/2-i exp(az/2)Wn , a(az) sin(w zd)dz = Jo

((a)p+w/2(r(i/2 ± a - n))

)

2 ^1

(1/2 ± a + p + 0/2,29), (n - p - 9/2, -29) (3/2,1);

; /-M Ua2)V

(34)

Corollary 5. If we take m = 1, p = 1, A = 0 and v = 1/2 in (5) and then by using (8), we obtain:

r

/ zp-d/2-1 exp(-az/2)Mn , a(az) sin(w zd)dz = o

(

w/2y/n(1/a)d/2r(2a + 1)

)

2^2

(a)P(r(a + n + 1/2))

(a + p + 9/2, 29), (n - p - 9/2, -29) (3/2,1), (a - 9 + 1/2, -29);

V4(a2)V

(35)

Corollary 6. If we take m = 1, p = 1, A = 0 and v = 1/2 in (6) and then by using (8), we obtain:

w/2^(1/a)e)2

r+tt /

J zp-e/2-1Wvaa(az)W-naa(az) sin(w zd)dz = i

2^2

(a)P

(p+^2±i ± a,9), (p + 9/2 + 1,29); x ( -w

(3/2,1), 2(1 + f+f2 ± n, 9);

G(a2)0

• (36)

Corollary 7. If we take m = 1,p = 1, A = 0 and v = -1/2 in (1) and then by using (9), we derive the following integral formula:

J±° zp-w/2-1 Jv(az)cos(z)dz = ^(4) (2/a)

p-w/2

1^2

(n+^^w); (—)(4)'

(1/2,1), ((

2+n-(p-w)2)

2

), -w);

(37)

X

X

Corollary 8. If we take m = 1, f = 1, A = 0 and v = -1/2 in (2) and then by using (9), we obtain:

p-w/2

zp-w/2-1Kv(az) cos(z)dz = 1/4^^

" (i)(a)

(1/2,1);

(38)

CoRoLLARY 9. If we take m = 1, f = 1, A = 0 and v = -1/2 in (3) and then by using (9), we obtain:

J zp-w/2-1 exp(-az)Kn(az) cos(b zw)dz = i

2 an

1^2

J™ \ (a)p-w/2 (p - w/2 ± n, 2w); ( -b2

(1/2,1), (p - w/2 + 1/2, 2w); V4(4a2)

\

; \4(4a2)wy

(39)

Corollary 10. If we take m = 1,f = 1, A = 0 and v = -1/2 in (4) and then by using (9), we obtain:

r

/ zp-d/2-1 exp(az/2) WV:a(az) cos(w z6)dz = ./0

(

n

0

2

(a)p-0/2(r(1/2 ± a - n)) (1/2 ± a + p - 0/2, 20), (n - p + 0/2, -20); ( -w2 (1/2,1); \4(a2)'

(40)

Corollary 11. If we take m = 1,f = 1, A = 0 and v = -1/2 in (5) and then by using (9), we obtain:

r

/ zp-d/2-1 exp(-az/2)Mn,a(az) cos(w z6)dz = 0

/yn(1/a)-6/2r(2a + 1) v (a)p(r(a + n + 1/2))

)

2 ^2

(a + p - 0/2 + 1/2, 20), (n - p + 0/2, -20) (1/2,1), (a - p + 0 + 1/2, -20);

Wa2W

(41)

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o

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Corollary 12. If we take m = 1,( = 1, A = 0 and v = -1/2 in (6) and then by using (9), we obtain:

r+<x

/ zp-e)2-1wv, a(az)W-n, a(az) cos(w zd)dz = o

V (a)P ,

2^2

(p-^2±i ± a, 9), (p - 9/2 + 1,

(1/2,1), 2(1 + —^ ± n, 9); V4(a2)

29); / -w2 \

; Wa2W

(42)

Corollary 13. If we take m = 1 in (1) and then by using (10), we derive the following integral formula:

yP-1

Jn (azJ '1 (b zw )dz = (1/2) [b/^+ ^2/a)

p+w(v+2\)

2^3

(1, 1)( n+p+wu+2w\,w);

(A +1, 1), (v + A + 1, (), ((2+n-(p+wv+2w1)), -w);

-b2 ) ( 4

(W

(43)

4a

Corollary 14. If we take m = 1 in (2) and then by using (10), we obtain:

v+21 ( \ p+w(v+2\)

zp-1Kn(azJ 1 (b zw)dz = (1/^ (b/2j (2/a)

" (1 1) (p+wv+2w\±n

2^2

,w); ±\

1,();\ 4 J\a2)

(44)

(A + 1,1), (v + A + 1,(); V 4 Corollary 15. If we take m = 1 in (3) and then by using (10), we obtain:

JQ zP exP(-az)Kn (azJ \ (b z )dz = ^ (2a)p+w(v+2X) ) x

2^3

(1,1), (p + wv + 2wA ± n, 2w); (A + 1,1), (v + A + 1, (), (p + w(v + 2A) + 1/2, 2w);

-b2

(—)

\4(4a2)w J

(45)

v4(4a2)w

Corollary 16. If we take m = 1 in (4) and then by using (10), we obtain:

r

/ zp-1 exp(az/2)Wn, a(az) J^'A1(w zd)dz = Jo ' '

276

X

X

o

w

(

(w/2)

v+2X

(a)p+6(v+2X) (r(1/2 ± a - n)).

3^2

(1,1)(1/2 ± a + p + 0v + 20A, 20), (n - p - 0v - 20A, -20); (A + 1,1), (v + A + 1, f);

w

V4(a2)6;

(46)

,4(a2)6,

Corollary 17. If we take m = 1 in (5) and then by using (10), we obtain:

3^3

r+m

/ zp-1 exp(-az/2)Mv,a(az)J^¿(w z6)dz =

Jo ' '

((w/1)v+2X(1/a)6(v+2X)r(2a + 1)) V (a)p (r(a + n + 1/2)) (1,1), (a + p + 0v + 20A + 1/2, 20), (n - p - 0v - 20A, -20); (A + 1,1), (v + A + 1, f), (a - p - 0v - 20A + 1/2, -20);

w2

4(a2)6

(47)

Corollary 18. If we take m = 1 in (6) and then by using (10), we obtain:

f+m

zp-1Wv,a(az)W-v,a(az)J^(w z6)dz =

(w/2)v+2X(1/a)6(v+2X)

. w

3^3

(1,1), (p+6(v+22X)+1 ± a,0), (p + 0(v + 2A) + 1, 20); / -w2 \ (A + 1,1), (v + A + 1, f), 2(1 + p+6(v2+2X) ± n, 0); V4(a2)6J

.(48)

Corollary 19. If we take m = 1,f = 1 and A = 1/2 in (1) and then by

using (11), we derive the following integral formula:

( )( ) v+1 ( ) p+w(v+1)

f0+m zp-1Jv (az)Hv (b zw )dz = I 1/2 J lb/2) I 2/a I x

2^3

(1,1)( n+p+wv+w ,w);

(3/2,1), (v + 3/2,1), ((2+n-(p+wv+w)), -w)

.(49)

Corollary 20. If we take m = 1, f = 1 and A = 1/2 in (2) and then by using (11), we obtain:

r+m ( )( ) v+1 ( ) p+w(v+1)

J zp-1Kv (az)Hv (b zw )dz = h/4) \b/2) i 2/aJ x

277

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cu c cu

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CC <

CD

S2 ■O

x

LU I—

o

o

2^2

(1, 1), (p+wv±w±n, w); i-b^f ±\

(3/2,1), (v + 3/2,(); 4 ) [a2)

• (50)

Corollary 21. If we take m = 1, ( = 1 and A = 1/2 in (3) and then by using (11), we obtain:

f+tt

/ zp-1 exp(-az)Kn(az)Hv(b zw)dz = o

f2ay/n(b/2)

V+1

2^3

(2a)(+w(v+1)

(1,1), (p + wv + w ± n, 2w); / -b2

_ (3/2,1), (v + 3/2,1), (p + w(v + 1) + 1/2, 2w); V4(4a2)

i— ^

; \4(4a2)wJ

• (51)

Corollary 22. If we take m = 1, ( = 1 and A = 1/2 in (4) and then by using (11), we obtain:

r +tt /

J zp-1 exp(az/2)Wn , a(az)Hv(w zd)dz =

(w/2)

V+1

3^2

(a)p+0(v+1)(r(1/2 ± a - n)) (1,1)(1/2 ± a + p + 9v + 9,29), (n - p - 9v - 9, -29); (3/2,1), (v + 3/2,1);

)

G(a2)0)

(52)

Corollary 23. If we take m = 1, ( = 1 and A = 1/2 in (5) and then by using (11), we obtain:

/+tt

zp-1 exp(-az/2)Mn a a(az)Hv(w zd)dz = " )

3^3

i (w/2)v+1(1/a)0(v+1) r(2a + 1)' V (a)P(r(a + n + 1/2)) ' X (1,1), (a + p + 9v + 9 + 1/2,29), (n - p - 9v - 9, -29); (3/2,1), (v + 3/2,1), (a - p - 9v - 9 + 1/2, -29); w2

G(a2)0)

(53)

Corollary 24. If we take m = 1, ( = 1 and A = 1/2 in (6) and then by using (11), we obtain:

j0+°° zp-1Wn ' a(az)W-n'a(az)Hv(w z0)dz = (^^¿lO^l^ x

278

X

3^3

(1,1), (p+6(v+1)+1 ± a, 0), (p + 0(v + 1) + 1, 20); ( -w2 ) (3/2,1), (v + 3/2,1), 2(1 + p+6(2;+1) ± n, 0); V4(a2)6J

. (54)

References

cn CD <M

S± p

O

t3

Chaurasia, V.B.L. & Pandey, S.C. 2010. On the fractional calculus of generalized Mittag-Leffler function. SCIENTIA Series A: Mathematical Sciences, 20, pp.113-122 [online]. Available at: https://citeseerx.ist.psu.edu/viewdoc/downlo ad?doi=10.1.1.399.5089&rep=rep1&type=pdf [Accessed: 9 February 2022].

Choi, J. & Agarwal, P. 2013. Certain unified integrals associated with Bessel e functions. Boundary Value Problems, art.number:95. Available at: |

https://doi.org/10.1186/1687-2770-2013-95.

Choi, J., Mathur, S. & Purohit, S.D. 2014. Certain new integral formulas involving the generalized Bessel functions. Bulletin of the Korean Mathematical Society, 51(4), pp.995-1003. Available at: https://doi.org/10.4134/BKMS.2014.51A995.

Jain, S., Choi, J., Agarwal, P. & Nisar, K.S. 2016. Integrals involving Laguerre type plynomials and Bessel functions. Far East Journal of Mathematical Sciences (FJMS), 100(6), pp.965-976. Available at: https://doi.org/10.17654/MS100060965.

Kachhia, K.B. & Prajapati, J.C. 2016. On generalized fractional kinetic equations involving generalized Lommel-Wright functions. Alexandria Engineering Journal, 55(3), pp.2953-2957. Available at: https://doi.org/10.1016/j.aej.2016.04.038.

Kiryakova, V.S. 2000. Multiple(multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. Journal of Computational and Applied Mathematics, 118(1-2), pp.241-259. Available at: https://doi.org/10.1016/S0377-0427(00)00292-2.

Mathai, A.M., Saxena, R.K. & Haubold, H.J. 2010. The H-function: Theory and Applications. New York, NY: Springer. Available at: https://doi.org/10.1007/978-1-4419-0916-9.

Mathai, A.M. & Saxena, R.K. 1973. Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Berlin Heidelberg: Springer- § Verlag. Available at: https://doi.org/10.1007/BFb0060468. §.

Menaria, N., Nisar, K.S. & Purohit, S.D. 2016. On a new class of integrals involving product of generalized Bessel function of the first kind and general class of polynomials. Acta Universitatis Apulensis, 46, pp.97-105 [online]. Available at: https://www.emis.de/journals/AUA/pdf/74_1366_aua_2841701 .pdf [Accessed: 9 February 2022].

Mondal, S.R. & Nisar, K.S. 2017. Certain unified integral formulas involving the generalized modified k-Bessel function of first kind. Communications of the Korean Mathematical Society, 32(1), pp.47-53. Available at: https://doi.org/10.4134/CKMS.c160017.

CD

"O

CD

.N

"TO

CD c CD OT

>

"O

CO <

ro

CD OT

<л

Paneva-Konovska, J. 2007. Theorems on the convergence of series in generalized Lommel-Wright functions. Fractional Calculus and Applied Analysis, 10(1), pp.59-74 [online]. Available at: http://eudml.org/doc/11298 [Accessed: 9 February 2022].

> Rainville, E.D. 1960. Special Functions. New York: The Macmillan Company.

В Srivastava, H.M. & Daoust, M.C. 1969. Certain generalized Neumann expan-

° sions associated with Kampe-de-Feriet function. Proceedings of the Koninklijke ^ Nederlandse Akademie van Wetenschappen Series A-Mathematical Sciences, E 72(5), pp.449-457.

о Srivastava, H.M. & Manocha, H.L. 1984. A treatise on generating functions.

Chichester, West Sussex, England: E. Horwood & New York: Halsted Press. ISBN: 3 9780853125082.

_ Whittaker, E.T. & Watson, G.N. 2013. A Course of Modern Analysis, reprint of

^ the fourth (1927) edition. Cambridge University Press, Cambridge Mathematical Library. Available at: https://doi.org/10.1017/CB09780511608759.

<c

ИССЛЕДОВАНИЕ ИНТЕГРАЛЬНЫХ ПРЕОБРАЗОВАНИЙ ОБОБЩЕННЫХ ФУНКЦИЙ ЛОММЕЛЯ-РАЙТА

Мохаммад Саид Кхан3, Сиразул Хакб, Мохаррам Али Кханв, Никола Фабианог

CD

_ a Университет медицинских наук Сефако Макгато,

о кафедра математики и прикладной математики,

г. Га-Ранкува, Южно-Африканская Республика

ш б Дж. С. Университет, департамент прикладных наук,

О г. Шикохабад, Фирозабад, штат Уттар-Прадеш, Республика Индия

g в Университет Умару Мусы Яр' Адуа, Кафедра математики и

статистики, г Катсина, Федеративная Республика Нигерия г Белградский университет, Институт ядерных исследований «Винча» - Институт государственного значения для Республики Сербия, г. Белград, Республика Сербия, корреспондент

РУБРИКА ГРНТИ: 27.23.21 Интегральные преобразования.

Операционное исчисление, 27.23.25 Специальные функции, 27.27.19 Функции многих комплексных переменных ВИД СТАТЬИ: оригинальная научная статья

Резюме: га

с

Введение/цель: Целью данной статьи является устано- со вление интегральных преобразований обобщенной функ- о. ции Ломмеля-Райта. а

о

Методы: Эти преобразования выражаются в терминах £ гипергеометрической функции Райта.

Результаты: В результате получены интегралы с тригонометрическими, обобщенными функциями Бесселя и Струве. ¡=

Е

Выводы: Вследствие применения данного метода получа- р ются различные интересные преобразования.

СТУДША О ИНТЕГРАЛНИМ ТРАНСФОРМАЦШАМА ГЕНЕРАЛИЗОВАНЕ ФУНКЦШЕ ЛОМЕЛА И РАJТА

ОБЛАСТ: математика

ВРСТА ЧЛАНКА: оригинални научни рад

Сажетак:

Увод/цил>: Циъ овог рада }есте успоставъаъе интеграл-них трансформаци]а генерализоване функци]е Ломела и Ра]та.

"О ф

N

Ключевые слова: обобщенные функции Ломмеля-Райта 2 и(2), преобразование Ханкеля, К-преобразование, функция ^ Райта, функция Уиттекера.

о

(Л

Е о

ч—

(Я

> "О

Мохамед Саид Кана, СиразулХакб, Мохарам Али Канв, Никола Фабианог

а Универзитет здравствених наука Сефако Макгато, ~

Департман за математику и приме^ену математику, Га-Ранкува, Република иужна Африка

б Универзитет и. С., Оде^е^е за приме^ене науке, Шикохабад

Фирозабад, Утар Прадеш, Република Инди]а в Универзитет Умару Муса Иарадуа, Департман за математику и

статистику, Катсина, Савезна Република Нигери]а г Универзитет у Београду, Институт за нуклеарне науке "Винча"- от

Институт од националног знача]а за Републику Срби]у, Београд Република Срби]а, ауторза преписку

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2

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Методе: Интегралне трансформаци]е изражене су помогу Ра}тове хипергеометри]ске функци]е.

Резултати: Доби}ени су интеграли ко\и укя>учу]у тригоно-метри]ске, генерализоване Беселове и Струвеове функци-¡е.

Закъучак: Као последице ове методе доби]а]у се разне за-нимъиве трансформаци]е.

ш

^ Къучне речи: генерализоване функци]е Ломела и Ра]та

о J(z), Ханкелова трансформаци]а, К-трансформаци]а, Ра]-

° това функци]а, Витакерова функци]а.

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0 Paper received on / Дата получения работы / Датум приема чланка: 10.02.2022. Manuscript corrections submitted on / Дата получения исправленной версии работы /

> Датум достав^а^а исправки рукописа: 14.03.2022.

¡< Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 16.03.2022.

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_ © 2022 Авторы. Опубликовано в "Военно-техническии вестник / Vojnotehnicki glasnik / Military ^ Technical Courier" (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией "Creative Commons"

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